Present Proof Research Articles

Transformations of triangle ladder diagrams

It is shown how dual space diagrammatic representation of momentum integrals corresponding to triangle ladder diagrams with an arbitrary number of rungs can be transformed to half-diamonds. In paper arXiv:0803.3420 [hep-th] the half-diamonds were related by conformal integral substitution to the diamonds which represent the dual space image of four-point ladder integrals in the four-dimensional momentum space. Acting in the way described in the present paper we do not need to use the known result for diamond (four-point) diagrams as an external input in deriving relations of arXiv:0803.3420 [hep-th], however, that result for the diamond diagram arises in the present proof as an intermediate consequence in a step-by-step diagrammatic transformation from the triangle ladder diagram to the half-diamond diagrams.

A Convergent Online Single Time Scale Actor Critic Algorithm

Actor-Critic based approaches were among the first to address reinforcement learning in a general setting. Recently, these algorithms have gained renewed interest due to their generality, good convergence properties, and possible biological relevance. In this paper, we introduce an online temporal difference based actor-critic algorithm which is proved to converge to a neighborhood of a local maximum of the average reward. Linear function approximation is used by the critic in order estimate the value function, and the temporal difference signal, which is passed from the critic to the actor. The main distinguishing feature of the present convergence proof is that both the actor and the critic operate on a similar time scale, while in most current convergence proofs they are required to have very different time scales in order to converge. Moreover, the same temporal difference signal is used to update the parameters of both the actor and the critic. A limitation of the proposed approach, compared to results available for two time scale convergence, is that convergence is guaranteed only to a neighborhood of an optimal value, rather to an optimal value itself. The single time scale and identical temporal difference signal used by the actor and the critic, may provide a step towards constructing more biologically realistic models of reinforcement learning in the brain.

Abel’s method on summation by parts and balanced q-series identities

The Abel method on summation by parts is reformulated to present new and elementary proofs of several classical identities of terminating balanced basic hypergeometric series. The examples strengthen our conviction that as traditional analytical instrument, the revised Abel method on summation by parts is indeed a very natural choice for working with basic hypergeometric series.

Toeplitz matrices with variable coefficients, pseudodifferential operators, and Strichartz's method

AbstractWe present simple proofs of sharp uniform boundedness results for sequences of Toeplitz matrices with variable coefficients. These proofs are based on Strichartz's method which reduces the problem of boundedness of operators “with variable coefficients” to the same problem for operators that have “constant coefficients” but act on Hilbert space valued functions. The paper also contains counterexamples showing the optimality of the boundedness results obtained with the help of Strichartz's method and a brief discussion of similar results for pseudodifferential operators (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Bailey's well-poised 6[formula omitted]-series implies the Askey–Wilson integral

Bailey's fundamental identity of bilateral well-poised ψ 6 6 -series is utilized to present shorter proofs for the two important q-beta integrals discovered by Askey and Wilson (1985) [4] and Askey (1987) [3]. Another rather general q-beta integral containing many extra parameters is similarly derived from Chu's extended Karlsson–Minton-type identity for a bilateral well-poised ψ 2 n + 6 6 + 2 n -series.

“A note on the economic lot size of the integrated vendor–buyer inventory system derived without derivatives”: A comment

Wee and Chung [Wee, H.M., Chung, C.T., 2007. A note on the economic lot size of the integrated vendor–buyer inventory system derived without derivatives. European Journal of Operational Research 177, 1289–1293] use the complete squares method to locate the optimal solution of the integrated system’s total cost TC ( Q , B , n ). However, their algebra method has shortcomings such that it may be invalid. The main purpose of this paper is to overcome those shortcomings and present complete proofs for Wee and Chung (2007).

Designing Document SQL (DSQL)

This article presents DSQL, a conservative extension of SQL, as an ad-hoc query language for XML. The development of DSQL follows the theoretical foundations of first order logic, and uses common query semantics already accepted for SQL. DSQL represents a core subset of XQuery that lends well to optimization techniques, while at the same time allows easy integration into current databases and applications that useSQL. The intent of DSQL is not to replace XQuery, the current W3C recommended XML query language, but to serve as an ad-hoc querying frontend to XQuery. Further, the authors present proofs for important query language properties such as complexity and closure. An empirical study comparing DSQL and XQuery for the purpose of ad-hoc querying demonstrates that users perform better with DSQL for both flat and tree structures, in terms of both accuracy and efficiency.

Graphene grown on Co(0001) films and islands: Electronic structure and its precise magnetization dependence

We achieve epitaxial growth of graphene on the ferromagnetic substrate Co(0001) by chemical vapor deposition using propylene. Structural and electronic properties are revealed by means of low-energy electron diffraction, scanning tunneling microscopy, and spin- and angle-resolved photoelectron spectroscopy. The results are compared to those of graphene/Ni(111) for which a magnetization-dependent Rashba effect has recently been reported. Bulklike Co(0001) films on W(110) can be separated into islands by heating, and subsequently they can be covered by graphene. Such system is used to present unambiguous proof that in graphene/Co(0001) there is no Rashba effect and no intrinsic dependence of the band dispersion on the magnetization occurs. The high spin polarization of secondary electrons reported for graphene/Ni(111) is confirmed for graphene/Co(0001) with higher values $(\ensuremath{\sim}25%)$ due to the larger magnetic moment of Co.

Equivalence of control systems with linear systems on Lie groups and homogeneous spaces

The aim of this paper is to prove that a control affine system on a manifold is equivalent by diffeomorphism to a linear system on a Lie group or a homogeneous space if and only if the vector fields of the system are complete and generate a finite dimensional Lie algebra. A vector field on a connected Lie group is linear if its flow is a one parameter group of automorphisms. An affine vector field is obtained by adding a left invariant one. Its projection on a homogeneous space, whenever it exists, is still called affine. Affine vector fields on homogeneous spaces can be characterized by their Lie brackets with the projections of right invariant vector fields. A linear system on a homogeneous space is a system whose drift part is affine and whose controlled part is invariant. The main result is based on a general theorem on finite dimensional algebras generated by complete vector fields, closely related to a theorem of Palais, and which has its own interest. The present proof makes use of geometric control theory arguments.

Griffiths Inequalities for Ising Spin Glasses on the Nishimori Line

The Griffiths inequalities for Ising spin glasses are proved on the Nishimori line with various bond randomness which includes Gaussian and $\pm J$ bond randomness. The proof for Ising systems with Gaussian bond randomness has already been carried out by Morita et al, which uses not only the gauge theory but also the properties of the Gaussian distribution, so that it cannot be directly applied to the systems with other bond randomness. The present proof essentially uses only the gauge theory, so that it does not depend on the detail properties of the probability distribution of random interactions. Thus, the results obtained from the inequalities for Ising systems with Gaussian bond randomness do also hold for those with various bond randomness, especially with $\pm J$ bond randomness.

Conditional proof of the Boltzmann-Sinai ergodic hypothesis

We consider the system of $N$ ($\ge2$) elastically colliding hard balls of masses $m_1,...,m_N$ and radius $r$ on the flat unit torus $\Bbb T^\nu$, $\nu\ge2$. We prove the so called Boltzmann-Sinai Ergodic Hypothesis, i. e. the full hyperbolicity and ergodicity of such systems for every selection $(m_1,...,m_N;r)$ of the external geometric parameters. The present proof does not use the formerly developed, rather involved algebraic techniques, instead it employs exclusively dynamical methods and tools from geometric analysis.

Panoptes: An Exploration Tool for Formal Proofs

Proof assistants aid the user in proving mathematical theorems by taking care of low-level reasoning details. Their user interfaces often present proof information as text, which becomes increasingly difficult to comprehend as it grows in size. Panoptes is a software tool that enables users to explore graphical representations of the formal proofs produced by the imps Interactive Mathematical Proof System. Panoptes automatically displays an imps deduction graph as a visual graph that can be easily manipulated by the user. Its facilities include target zooming, floating information boxes, node relabeling, and proper substructure collapsing.

Generalized demonstration of Ramo's theorem with space charge and polarization effects

Ramo's theorem provides a convenient way of simulating the currents induced through the electrodes of a radiation detector by the movement of charge carriers. The concept of weighting potential or of weighting field is the key ingredient in the calculation of the induced currents. A new demonstration of the theorem, based on energy balance, is provided that explicitly takes into account the effect of material polarization. It is shown that Ramo's theorem is valid in an arbitrary linear material and that polarization charges at interfaces between different materials or in the bulk of inhomogeneous materials must be included in calculating the weighting potentials. The use of our definition of weighting potential is also demonstrated for non-linear materials. The present proof is thus more generally valid than previous ones.

On Saint-Venant’s Principle for Elasto-Plastic Bodies

This paper concerns Zanaboni’s version of Saint-Venant’s principle, which states that an elongated body in equilibrium subject to a self-equilibrated load on a small part of its smooth but otherwise arbitrary surface, possesses a stored energy that in regions of the body remote from the load surface decreases with increasing distance from the load surface. We here prove this formulation of Saint-Venant’s principle for elastic-plastic bodies. The present proof, which for linear elasticity considerably simplifies that developed by Zanaboni, depends crucially upon the principle of minimum strain energy to obtain a fundamental inequality that leads to the required result. Differential inequalities are not involved. The conclusion is not restricted to cylinders but is valid for plastic bodies of general geometries. Although no conditions are imposed on the plastic theories discussed here, counter-examples indicate that in certain circumstances the fundamental inequality, and hence the result, may be valid only for restricted data that includes the body’s minimum length.

A complete proof on the solution procedure for non-instantaneous deteriorating items with permissible delay in payment

Ouyang et al. [Ouyang, L. Y., Wu, K. S., & Yang C. T. (2006). A study on an inventory model for non-instantaneous deteriorating items with permissible delay in payments. Computers & Industrial Engineering, 51, 637–651] present an inventory model for non-instantaneous deteriorating items with permissible delay in payments. They develop some theorems to characterize the optimal solutions and provide an easy-to-use method to find the optimal replenishment cycle time. Although, their inventory models are correct and interesting, processes of arguments to derive those theorems and the easy-to-use method to search for the optimal replenishment cycle time are not complete. So, the main purpose of this paper is to overcome those shortcomings and present complete proofs for Ouyang et al. (2006).

Common fixed points theorems for fuzzy mappings

Recently, Abu-Donia [Abu-Donia HM. Common fixed points theorems for fuzzy mappings in metric space under ϕ contraction condition. Chaos, Solitons & Fractals 2007;34:538–43] studied the Hausdorff metric between fuzzy subsets via its correspondence between the two classical sets and established two common fixed point theorems for two fuzzy mappings which are useful in geometric problems arising in high energy physics. However, the proofs of Abu-Donia’s main results are incorrect. The purpose of this paper is to present correct proofs of these results.

Comments on the EOQ model under retailer partial trade credit policy in the supply chain

Huang and Hsu [2007. An EOQ model under retailer partial trade credit policy in supply chain. International Journal of Production Economics, doi:10.1016/j.ijpe.2007.05.014] investigate the inventory system as a cost minimization problem to determine the retailer's optimal inventory policy under the supply chain. They develop two easy-to-use theorems to locate the optimal inventory policy for the retailer. Although their inventory models are correct and interesting, processes of arguments to derive those theorems to find the optimal inventory policy are not complete. So the main purpose of this paper is to overcome these shortcomings and present complete proofs for Huang and Hsu (2007).

Uniform Uncertainty Principle for Bernoulli and Subgaussian Ensembles

The paper considers random matrices with independent subgaussian columns and provides a new elementary proof of the Uniform Uncertainty Principle for such matrices. The Principle was introduced by Candes, Romberg and Tao in 2004; for subgaussian random matrices it was carlier proved by the present authors, as a consequence of a general result based on a generic chaining method of Talagrand. The present proof combines a simple measure concentration and a covering argument, which are standard tools of high-dimensional convexity.

On simultaneous triangularization of collections of compact operators

In this paper we consider collections of compact (resp. C p class) operators on arbitrary Banach (resp. Hilbert) spaces. For a subring R of reals, it is proved that an R-algebra of compact operators with spectra in R on an arbitrary Banach space is triangularizable if and only if every member of the algebra is triangularizable. It is proved that every triangularizability result on certain collections, e.g., semigroups, of compact operators on a complex Banach (resp. Hilbert) space gives rise to its counterpart on a real Banach (resp. Hilbert) space. We use our main results to present new proofs as well as extensions of certain classical theorems (e.g., those due to Kolchin, McCoy, and others) on arbitrary Banach (resp. Hilbert) spaces.

Tissue Engineering of Skeletal Muscle

Loss of skeletal muscle profoundly affects the health and well-being of patients, and there currently is no way to replace lost muscle. We believe that a key step in the development of a prosthesis for reconstruction of dysfunctional muscular tissue is the ability to reconstitute the in vivo-like 3-dimensional (3D) organization of skeletal muscle in vitro with isolated satellite cells. In our present proof of principle studies, we have successfully constructed a multilayered culture of skeletal muscle cells, derived from neonatal satellite cells, that are distributed in a 3D pattern of organization that mimics many of the features of intact tissue. These multilayered cultures are composed of elongated multinucleated myotubes that are MyoD positive. Histological studies indicate that the multiple layers of myotubes can be distinguished. Expression of muscle-specific markers such as myosin heavy chain, dystrophin, integrin alpha-7, alpha-enolase, and beta-enolase was detected using real-time reverse transcriptase polymerase chain reaction at levels near adult values. Physiological measurements of the engineered skeletal muscle showed that they tetanize and display physiologic force length behavior, although developed force per cross-sectional area was below that of native rat skeletal muscle.